Comment on ``Proposed central limit behavior in deterministic dynamical systems''

TL;DR

This paper critiques Grassberger's recent simulation results on the logistic map, clarifying errors and emphasizing the irrelevance of Levy distributions and the non-universality of transient-focused sums.

Contribution

It identifies errors in Grassberger's simulations and clarifies the conditions under which previous results remain valid.

Findings

Grassberger's simulations were performed outside the relevant parameter region.

Levy distributions are not applicable to this problem.

Distributions of transient sums are unlikely to be universal.

Abstract

In a recent Brief Report [Phys. Rev. E 79 (2009) 057201], Grassberger re-investigates probability densities of sums of iterates of the logistic map near the critical point and claims that his simulation results are inconsistent with previous results obtained by us [U. Tirnakli et al., Phys. Rev. E 75 (2007) 040106(R) and Phys Rev. E 79 (2009) 056209]. In this comment we point out several errors in Grassberger's paper. We clarify that Grassberger's numerical simulations were mainly performed in a parameter region that was explicitly excluded in our 2009 paper and that his number of iterations is insufficient for the region chosen. We also show that, contrary to what is claimed by the author, (i) Levy distributions are irrelevant for this problem, and that (ii) the probability distributions of sums that focus on transients are unlikely to be universal.